“Why is geometry often described as ‘cold’ and ‘dry?’ One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. Nature exhibits not simply a higher degree but an altogether different level of complexity.” – Benoit Mandelbrot
If you thought, even for a second, that any of the mathematical discoveries I have presented to you on this website have been absolutely enlightening, then you have been disillusioned; those pieces of information have been known for centuries. Today, we as mathematicians will break with the obvious, with the pieces of information that we have known to be true for a long time. Rather than concern ourselves with numbers, special constants, and sets, all of which are archaic, we will observe mathematics of the twenty-first century—our generation’s contributions to mathematics. Today, we will study, more likely peruse, fractal geometry, an idea which was established a mere thirty-five years ago by Benoit Mandelbrot. Have a paper and pencil handy: You will need it as we go through the very basics of this fascinating topic!
To get the ball rolling, I want to start with an activity. (It will require paper and a pencil.) Draw a vertical line segment on the right side of your paper. Erase the middle third of this segment. Replace this missing third with a “carrot” pointing to the right (i.e. draw two line segments to join the top third line segment and bottom third line segment in such a way that these two line segments you are drawing also make an arrow pointed to the right). You should now have a figure composed of four line segments. Erase the middle third in each of these line segments. Replace those missing thirds with a smaller carrot. You should now have a figure composed of sixteen line segments. Erase the middle third in each of these line segments. Replace those missing thirds with an even smaller carrot. You should now have a figure composed of sixty-four line segments.
Rather than continue to sketch the pattern, which is bound to get very messy, let us stop and think critically. What does this pattern resemble? Well, it should become increasingly obvious that this pattern looks like the edges of a snowflake (and just in time for that season to begin). As it turns out, the figure you have drawn is known as the von Koch snowflake. It is, perhaps, the third most famous fractal in all of mathematics. (Do not worry: I will show you the second and first most famous fractals, respectively, in the next blog.) For further purposes of discussion (and doubts in my ability to instruct effectively), I have posted the von Koch fractal below.
So, we have generated the fractal above quite simply, but how do we define fractals in general? As it turns out, the true mathematical definition is way beyond our comprehension (and unnecessary for this blog’s purposes). Figuratively, however, we can postulate a definition based on just this one example. Notice: No matter how closely we try to observe the snowflake, we always see the same figure. For instance, on the leftmost picture of the snowflake, if I chose to look at one of those tiny segments that comprise the entire figure under a microscope, the return in my microscope would be the exact same figure displayed by the leftmost snowflake at normal magnification. To put simply, we can never reach the fundamental components of a figure that is a fractal; we can only observe smaller and smaller “worlds” (patterns in the figure that continue for infinity). For our purposes, therein lies our definition of a fractal. It is a geometrical object comprised of infinitely many “worlds.”
Now, I want to shed light on the significance of our definition for a fractal. In standard mathematics, we focus primarily on principles of Euclidian geometry, a study which each one of us has experienced in some form at the grade-school level. For example, we have studied that two lines parallel to each other will never intersect or that three points in space will establish a plane. A direct corollary to this study comes from a completely different study in standard mathematics—the calculus. (I do realize at this point that some readers have not studied calculus, and so I will try to be as conceptual as possible.)
A major component of calculus deals with change. In effect, how does something change over an infinitely small interval? To somebody who has not studied calculus, this question seems to have an impossible answer, but right now, I will show you that it is not so impossible. Draw any generic curve on your paper. Cut that curve into smaller and smaller intervals. Notice: As you make smaller and smaller intervals, your curve becomes a collection of small lines! Therefore, over any given small interval, a curve can be represented by a line in that small interval. The change in the curve over that small interval is directly related to the steepness of the line that represents the curve in that small interval. This same process can be done for a surface, and the conclusion is that a surface becomes a collection of small planes. Now, how does this idea of small intervals of change relate to Euclidian geometry and fractals?
Imagine any object you want in three-dimensional space. Note the surface of that object over small intervals of area. Over these small intervals, the surface can be represented by a plane, correct? More generally, any surface of a three-dimensional object can be thought of as a collection of small planes, true? Because planes are defined by Euclidian geometry, we conclude that three-dimensional objects are Euclidian when their surface can be comprised everywhere of small planes. If we use the same logic, we can say something similar for two-dimensional objects: Two-dimensional objects are Euclidian if their curves can be comprised everywhere of small lines.
Doing all this explanation just to come up with these definitions probably seems tedious, but there is a method to my madness.
Consider the fractal you drew earlier. Is that figure Euclidian? Well, so far, we only know how to prove shapes to be Euclidian if they are of two or three dimensions. In this case, I will logically assume the von Koch snowflake to be a two-dimensional figure because it exists in the plane of my paper. With this assumption, what do all two-dimensional Euclidian figures have in common? Their curves can be broken into very small intervals, and when they are divided into these intervals, collections of small lines represent the curves. Here, we run into a problem. Recall our definition of a fractal: It is a figure where infinitely many worlds exist. The smaller we examine the figure, the more worlds we see (because the pattern imbedded within the figure reiterates itself to infinity). Therefore, the von Koch snowflake is not comprised of lines; rather, it is composed of worlds within worlds within worlds within worlds. Therefore, we come to the conclusion that it is not a Euclidian figure.
This is a very odd result. If you draw any other curve in the plane of your paper, it will surely be Euclidian. What makes this fractal so different? How can it seem to be a two-dimensional curve yet not be Euclidian? It defies standard mathematics!
Well, it turns out that the von Koch snowflake, and any other fractal for that matter, is not one-dimensional, two-dimensional, three-dimensional, or even four-dimensional. Fractals have fractional dimension (hence the name fractal). We will discuss fractional dimension next week and observe the strange life beyond Euclidian objects.
Welcome to the mathematics of the future, our generation’s contribution to mathematics. Be prepared for the freshest take on mathematics you have experienced since counting!